Are you wanting just aggregate counts of different calls, or the ability to interrogate a specific call?
Prometheus can help with the former, but logs & traces are the tools for specific details.
When you increment your counter (of number of API calls made) you can attach labels. Examples would be the name of the API call being made and the response code (which can then identify calls which have been rate limited). It is also common to include a label to identify what initiated the call (such as a subsystem name) - you’d then need to figure out how to obtain this. The simplest option would be to edit your code to pass a string to the code making the API call, but I’d imagine you could extract the caller details programmatically via stack traces, etc.